curve #4640

y² + xy = x³ − 830720957029924587169379982242644540x + 291156887021899292579682514661632669722940826017607696

a-invariants: [1, 0, 0, -830720957029924587169379982242644540, 291156887021899292579682514661632669722940826017607696]
c-invariants: [39874605937436380184130239147646937921, -251559550386920988848657601573805196916816232400683456225]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 29059786682435112110130040375597681105691029237985002427121492680864975748630737397754638
discriminant (Δ): 68163181528330024722196160644746065547503759067289529338336187640174381096708114484338193528317564108275712
Faltings height: 19.4155
naive height: 259.7364
primes of bad reduction: 2, 3, 7, 13, 23, 379, 136999, 73127333383811939, 4082787399006094054822683523, 149272165591023108979543678416653
regulator: 12112408874014264920740345.917576481388700879346335118153721439379592750648907006914422
submitted by: apkorte
submitted at: 2026-10-19 11:22:48 UTC
last updated: 2026-10-19 11:22:48 UTC

Witness: 24 independent points log in to add more points →

(541443564844677752, 9928749675785360147451524)
(-765018799279764448, -692057731941087470143141156)
(25453279823258171264/25, 88455473177341094514816870028/125)
(5199995129446287752, -11686702568233738491529377676)
(639522460907662136, 146457549682523800041182276)
(-821861528782742248, 647118904349879314688821124)
(-143279751022245256, -638154351502053334270646524)
(546602846743772816, -19837675216545015210416404)
(67412927998489751120/121, 47380664181052243347426956356/1331)
(120679563640204656572/289, 642087091771463862943361793232/4913)
(10537906119091974632/49, 120025112383148843409028439132/343)
(3159049832488367596/9, -5586771013699389255090631040/27)
(4280878840504713080, 8670986094792399301847675396)
(543408794439138218, -14166525495467214761040826)
(469020803544754898, 68606950934600252896750814)
(-804187761894364630, 662668732345795018233552326)
(-49774381093759258408/49, -101078555986167524910298871236/343)
(12109624330970752424/25, 6147667979027567627849931508/125)
(-665583167105333896, -741091624027873951728718204)
(498312758088089204, 30608326452837316131217496)
(708691750410445880, -241594651204758533680114876)
(19623521958056152952, -86837011808883967936473461116)
(-791696598225603448, 672765542854673726394156404)
(426445019936363498, -120213903071386712953274362)

Commentary

Specialization at T = 653/80 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -5295620710700377671600T^8 - 94037966474871021583680T^7 - 590731768793188987891776T^6 - 316565515830122546981184T^5 - 762681779746720476850440T^4 + 1987613918828860347667800T^3 + 1569564836467837283726676T^2 - 3214905333565214256563376T - 2212765344524109268510452, A6 = 149936157543712438778032606320000T^12 + 4003077612324280274862537287520000T^11 + 40256463697048454693428799356032000T^10 + 181374780086403872414844938171109120T^9 + 402662249843699366956864710038784096T^8 - 758031533540911762791576885234671328T^7 - 399143032260021392125750735091011632T^6 + 2151897782383821345209559268211414304T^5 + 557784900106071266465324558195950116T^4 - 3137783098558594528537444090769421600T^3 - 420721733859426366310009677118678488T^2 + 2767709433062054442394845582008506272T + 1268413949854822572813035476740281796. It is a 2-neighbour of another fibration built here: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-19 11:22:48 UTC · history